--- layout: default --- Retrospective Rating

Recall: Prospective WC Rating

2000 Q6

\(\begin{align} \text{Premium} = &\sum\limits_{i \in class} \dfrac{Payroll_i}{100} \text{Manual Rate}_i &\cdots (1)\\ &\times \: \text{Exp Mod} &\cdots (2)\\ &\times \: \text{Sch Mod} &\cdots (3)\\ &\times \: (1 - \text{Premium Discount %}) &\cdots (4) \end{align}\)

\((1)\) Manual Premium

\((1) \times (2)\) Modified Premium

\((1) \times (2) \times (3)\) Standard Premium

\((1) \times (2) \times (3) \times (4)\) Guaranteed Cost Premium

Retrospective Rating: Intro

Uses insured’s loss experience during the current policy term to determine their current policy premium

Initial premium collected at the start (deposit premium) \(\Rightarrow\) Adjust based on reported loss starting 6 months after policy expires, then every 12 months after

\(\text{Premium} = \left[\text{U/W Expense & Profit xTax} + \left(1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\right) \times Loss_{actual}\right] \times \text{Tax Multiplier}\)

There’s always going to be a max and min premium or else this would defeat the purpose of risk transfer

Caps are applied to the premium amount \(\equiv\) Capping actual losses at max/min aggregate loss limits \(\Rightarrow\) Add a charge to the premium calculation to account for this

In addition to aggregate caps, can also limit losses per occurrence (Use limited losses in the retro premium calculation) \(\Rightarrow\) A separate charge to the premium calculation to account for the per occurrence limit

With both occurrence and agg limits, need to consider the overlap between the 2 charges

Retrospective Rating: w/o Occurence Limit Important Formulas

NCCI retro premium formula for each individual policy

\(R = (b + C \times A + P \times C \times V) \times T\) Memorize

For a balanced plan:

\(\mathrm{E}[R] = \text{Premium for Prospectively Rated Policy} = b = e - (C - 1)\mathrm{E}[A] + C \times I\) Memorize

2001 Q32

Understanding the Net Insurance Charge: w/o Net Insurance Charge

Ignore the retro development for now

\(R = (b + C \times A) \times T \: \: \: \text{for} \: H \leq R \leq G\)

Max and min premiums \(\equiv\) Aggregate limits on losses

Formula
2001 Q31

\(\begin{align} H &= (b + C \times A_H) \times T &= \text{min}\\ G &= (b + C \times A_G) \times T &= \text{max}\\ \end{align}\)

\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)

Understand Formula

\(\begin{array}{ccc} \text{Net Insurance Charge} = &I = (&\text{Ins Chg} &- &\text{Ins Saving}&) \times \mathrm{E}[A]\\ &I = (&\dfrac{\mathrm{E}[Loss > A_G]}{\mathrm{E}[A]} &- &\dfrac{\mathrm{E}[Loss < A_H]}{\mathrm{E}[A]}&) \times \mathrm{E}[A]\\ &I = (&\text{Tbl M Max Prem Chg} &- &\text{Tbl M Min Prem Savings}&) \times \mathrm{E}[A]\\ &I = (&\phi(r_G) &- &\psi(r_H)&) \times \mathrm{E}[A]\\ &I = (&\phi(\dfrac{A_G}{\mathrm{E}[A]}) &- &\psi(\dfrac{A_H}{\mathrm{E}[A]})&) \times \mathrm{E}[A]\\ \end{array}\)

Table M

Rows are entry ratios = \(r = \dfrac{Loss_{Actual}}{Loss_{Expected}}\) Understand

  • \(\because\) Standardized \(\therefore\) Table M can be applied to various states and LOB

Columns are for different risk size groups

  • Since distn of charges vary by risk size \(\Rightarrow\) Curve differ by risk size

Entry ratio at maximum premium \(= r_G = \dfrac{A_G}{\mathrm{E}[A]}\)

Entry ratio at minimum premium \(= r_H = \dfrac{A_H}{\mathrm{E}[A]}\)

Retrospective Rating: with Occurence Limit

Per occurrence limit impact the likelihood of hitting the max/min premiums \(\Rightarrow\) Table M not appropriate \(\Rightarrow\) Need to recognize the potential overlap between the occurrence and aggregate limits

3 options to handle this:

Option 1: Limited Loss Table M Important Formulas

\(R^* = (b^{LLM} + C \times A^* + P \times C \times F + P \times C \times V) \times T\)Memorize

  • \(b^{LLM}\): From the limited loss Table M

  • \(A^*\): Reported limited loss @ t (might include ALAE)

  • \(F = ELF = \dfrac{\mathrm{E}[Loss \geq \text{Per Occ Limit}]}{\text{Standard Premium}}\)
    • Varies by State + HG + Occ Limit
    • Recall Robertson paper

  • \(A \neq A^* + PF\)

  • \(\mathrm{E}[A] = \mathrm{E}[A^*] = PF\)

  • \(PCF\) and \(PCV\) are electives

  • \(V\): Development different with occ limit than w/o

Theoretically correct approach

Caveat:

  • Requires a large number of tables

  • Charge need to vary by occ limit, entry ratio, risk size group, HG, etc

Option 2: Table M with ICRLL Adjustment (NCCI)

ICRLL = Insurance Charge Reflecting Loss Limitation

Change the Table M column used to be the column appropriate for a larger size risk

  • Per occurrence limit reduces the skewness of the entry ratio distn \(\Rightarrow\) Similarly for larger risk size

Caveat:

  • Table M doesn’t vary by loss limit \(\Rightarrow\) Compromises a bit on accuracy

Option 3: Table L (WCIRB)

\(R^* = (b^L + C \times A^* + P \times C \times V) \times T\)

Accurately correct for overlap between the occurrence and aggregate charges similar to the LLM

Combines occurrence charge with the aggregate charge

  • Whereas LLM’s occ charge has to be added separately

WCIRB uses different HG than NCCI

\(V\) varies by occ limit

  • \(\uparrow\) limits \(\uparrow\) \(V\) \(\because\) \(\uparrow\) potential for development

NCCI Filings for Optional Components

NCCI do full rates or loss costs depending on the state

Full Rates:

Loss Costs:

Constructing Table M

Aggregate distn across a large number of risks

Table M Charge

Revisit! This seems wrong…

2 Components to build Table M

Normalize table if the sample average \(\neq\) assumed expected value:

Constructing Table M Example

Annual Claim Total ($) # of Policies
1000 8
1500 3
4000 2
10000 5
15000 2
40000 1

Each risk has $10,000 of standard premium

\(\text{Max Premium} \Leftrightarrow\) 80% LR w.r.t. Standard Premium

\(\text{Min Premium} \Leftrightarrow\) 20% LR w.r.t. Standard Premium

Loss @ Max Premium = 80% \(\times\) $10,000 = $8,000 = \(A_G\)

Loss @ Min Premium = 20% \(\times\) $10,000 = $2,000 = \(A_H\)

Total # of Policies = 21

Total Losses = $140,500

\(\mathrm{E}[A] = \text{Expected Loss per Policy} = \dfrac{\text{Total Losses}}{\text{Total # of Policies}} =\) $6,690.48 = Total area under curve

# Transformation of the data to plot
d1_ecdf <-
  d1 %>%
  rbind(c(0,0),.) %>%
  mutate(`% Policies` = `# of Policies` / sum(`# of Policies`),
         `Claim EWCDF` = cumsum(`% Policies`))
# Note the addition of 0 row
Annual Claim Total ($) # of Policies % Policies Claim EWCDF
0 0 0.0000000 0.0000000
1000 8 0.3809524 0.3809524
1500 3 0.1428571 0.5238095
4000 2 0.0952381 0.6190476
10000 5 0.2380952 0.8571429
15000 2 0.0952381 0.9523810
40000 1 0.0476190 1.0000000

Note that the steps are drawn in a “vh” manner starting from origin

Table M Charge in dollars = Area under curve and above $8,000 line \(A_G\)

Table M Savings in dollars = Area under $8,000 line \(A_G\) and above curve

Net Insurance Charge = Difference of the Table M Charge in dollars and Table M Savings in dollars

3 Ways to Calculate the Areas

  1. Horizontal Slices (Table M Method)
    • Calculating the entry ratio first
  2. Vertical Slices
  3. Calculate area within limit and then subtract it from the expected loss
    • = Limiting losses at the max/min

Horizontal Slices (Table M Method)

Calculating the entry ratio first

  • Used to get the “height” of the segment

Starting from the top of the graph

Cumulate the horizontal length of each section from top to bottom

  • The % risks above \(r_i\) piece

Advantages:

  • Efficient when calculating multiple \(\phi{r_i}\)
  • Faster when there are many risks

Table M Construction from Sampled Loss

1. Start with table of sampled loss amount and respective frequency

Annual Claim Total ($) # of Policies
1000 8
1500 3
4000 2
10000 5
15000 2
40000 1

2. Add rows for 0 loss and entry ratio

tbl_m <-
  d1 %>%
  rbind(c(0,0), c(minlr * sp,0), c(maxlr *sp,0)) %>%
  arrange(`Annual Claim Total ($)`) %>%
  rename(`# risks` = `# of Policies`)

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks
0 0
1000 8
1500 3
2000 0
4000 2
8000 0
10000 5
15000 2
40000 1

3. Calculate Entry Ratio \(r\) * $

tbl_m <-
  tbl_m %>%
  mutate(`Entry Ratio (r)` = `Annual Claim Total ($)` / weighted.mean(`Annual Claim Total ($)`, `# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r)
0 0 0.0000000
1000 8 0.1494662
1500 3 0.2241993
2000 0 0.2989324
4000 2 0.5978648
8000 0 1.1957295
10000 5 1.4946619
15000 2 2.2419929
40000 1 5.9786477

4. Calculate # of risk above each row (Loss)

tbl_m <-
  tbl_m %>%
  mutate(`# risks above` = sum(`# risks`) - cumsum(`# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r) # risks above
0 0 0.0000000 21
1000 8 0.1494662 13
1500 3 0.2241993 10
2000 0 0.2989324 10
4000 2 0.5978648 8
8000 0 1.1957295 8
10000 5 1.4946619 3
15000 2 2.2419929 1
40000 1 5.9786477 0

5. Calculate % of risk above each row

tbl_m <-
  tbl_m %>%
  mutate(`% risks above` = `# risks above` / sum(`# risks`))

kable(tbl_m, align = 'l')
Annual Claim Total ($) # risks Entry Ratio (r) # risks above % risks above
0 0 0.0000000 21 1.0000000
1000 8 0.1494662 13 0.6190476
1500 3 0.2241993 10 0.4761905
2000 0 0.2989324 10 0.4761905
4000 2 0.5978648 8 0.3809524
8000 0 1.1957295 8 0.3809524
10000 5 1.4946619 3 0.1428571
15000 2 2.2419929 1 0.0476190
40000 1 5.9786477 0 0.0000000

6. Calculate the Table M Charge \(\phi(r)\)

\(\phi(r) = 0\) for the largest value of r \(\Rightarrow\) No \(\mathrm{E}[L]\) above the largest loss

\(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\) Memorize Formula

Savings \(= \psi(r) = \phi(r) + r - 1\) Memorize Formula

Vertical Slices

Looking at groups of losses based on the size of loss

Probably start from the right but don’t seem to matter

  • For Tbl M Charge: \(\text{ % of Policies hitting cap} \times \text{XS Loss Amount}\)
  • For Tbl M Savings: $

Advantages:

  • Natural, since that’s how the data is presented
  • Easier to understand
  • Faster if just looking at 1 \(r\)

Understanding Table M Important Concepts

alt text

Note that the y axis is now the entry ratios and not losses like it was in the earlier example

Defn:

\(Y = \dfrac{A}{\mathrm{E}[A]}\)

\(F(y) = \text{CDF of } y\)

Area under curve:

\(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A]} = 1\)

Table M Charge and Savings Formulas

\(\phi(r) = \int\limits_r^{\infty} (y - r) dF(y)\)

\(\psi(r) = \int\limits_r^{\infty} (r - y) dF(y)\)

Key Properties - Tbl M Charge Important

\(\phi(0) = 1, \: \phi(\infty) = 0\)

\(\phi(r)'(f) = -G(r) \leq 0\)

\(\phi''(r) = f(r)\)

Key Properties - Tbl M Savings Important

\(\psi(0) = 0, \: \psi(\infty) = \infty\)

\(\psi(r)'(f) = F(r) \geq 0\)

\(\psi''(r) = f(r) = \phi''(r)\)

Note from figure above, the area under \(r = \psi(r) + (1 - \phi(r)) \Rightarrow \psi(r) = \phi(r) + r -1\)

\(\phi\) is monotonically decreasing function of premium size

Table M Charges and Premium Size

Large sample and large risks \(\Rightarrow\) \(\downarrow\) variance in Loss Ratio and Entry Ratio than large sample but small risks

\(\lim\limits_{\text{Premium Size}\rightarrow \infty} Var(y) = 0\)

\(\phi(r) = \begin{cases} 1 - r &r \leq 1\\ 0 &r > 1 \end{cases}\)

CDF of Entry Ratio vs Entry Ratio

alt text

On the other hand, as \(\lim\limits_{\text{Premium Size} \rightarrow 0} \phi(r) = 1\)

Graphs of Entry Ratios vs Table M Charge/Savings as premium size increase

alt text

Net Insurance Charge - w/o Occurence Limit Important Concepts

\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)

\(A_H = r_H \mathrm{E}[A]\) and similarly for \(A_G\)

alt text

Blue area = \(\dfrac{\mathrm{E}[{\color{blue}L}]}{\mathrm{E}[A]}\)

\(\begin{array}{ll} \dfrac{\mathrm{E}[L]}{\mathrm{E}[A]} &= 1 + \psi(r_H) - \phi(r_G) \\ \mathrm{E}[L] &= \mathrm{E}[A] + [\psi(r_H) - \phi(r_G)]\mathrm{E}[A] \\ \mathrm{E}[L] &= \mathrm{E}[A] - I \end{array}\)

Memorize Understand

\(\begin{array}{lll} R &= (b + CL)T\\ \mathrm{E}[R] &= (b + C\mathrm{E}[L])T\\ \mathrm{E}[R] &= (b + C(\mathrm{E}[A] - I))T & (1)\\ \end{array}\)

Recall if the plan is balance \(\Rightarrow\) \(\mathrm{E}[R] = \text{Guaranteed Cost Premium}\)

\(\begin{array}{lll} \text{Guaranteed Cost Premium} &= (e + \mathrm{E}[A])T & (2)\\ &=(1-D)P & (3)\\ \end{array}\)

Combine \((1)\) and \((2)\)

\((b + C(\mathrm{E}[A] - I))T = (e + \mathrm{E}[A])T\)

\(b = e - (C - 1)\mathrm{E}[A] + CI\) Memorize Formula

2009 Q31, 2011 Q8

Question that ask for G or H usually use balance equation

Alt formula: \(b = (1 - D)\dfrac{P}{T} - C\mathrm{E}[A] + CI\)

Use this if we don’t’ have max/min premium

Table M Balance Equations

Basic premium and max/min premium depends on each other \(\Rightarrow\) Need trial and error to get the right Table M row

First balance equation

Value (change) difference

Retro premium formula with max loss:
\(H = (b + CA_H)T = (b + Cr_H\mathrm{E}[A])T\)

Start with the GCP formula and subtract \(H\)

\(\begin{array}{ll} \text{GCP} - {color{blue}H} &\\ (e + \mathrm{E}[A])T - {\color{blue}H} &= (b + C(\mathrm{E}[A] - I))T - {\color{blue}{(b + Cr_H\mathrm{E}[A])T}}\\ &= CT\{\mathrm{E}[A] - I - r_H\mathrm{E}[A]\}\\ &= CT\{\mathrm{E}[A] - \mathrm{E}[A](\phi(r_G) - \psi(r_H)) - r_H \mathrm{E}[A]\}\\ &= C\mathrm{E}[A]T(\psi(r_H) - r_H + 1 - \phi(r_G))\\ \end{array}\)

Move everything to one side and we get: Important Memorize

\(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)

Second balance equation

Entry difference, based \(G - H\)

Important Memorize

\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)

Can do all the above looking at them as a % of Standard Premium

Constructing a Limited Loss Table M

Same as Table M but with limited losses \(A^*\) for each policy

Entry Ratio: \(r^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)

Note that the denominator of the Entry Ratio is the \(\mathrm{E}[A^*]\), where as in Table L it’s not limited \(\mathrm{E}[A^*]\)

Understanding a Limited Loss Table M

Note that the y axis is now limited entry ratios \(y^*\)

alt text

Entry Ratio: \(Y^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)

\(F^*(Y^*) = \text{CDF of }Y^*\)

Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\) Know

Note that \(\mathrm{E}[Y^*] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A^*]}\) no surprise

But \(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A^*]} = 1 + k\dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)

Integration and key properties all the same as Table M but with limited loss

Same thing for what happens when \(\lim\limits_{\text{Premium Size} \rightarrow \infty}\) and \(\lim\limits_{\text{Premium Size} \rightarrow 0}\)

Net Insurance Charge - LLM

\(R^* = (b^{LLM} + C{\color{blue}{L^*}}+PCF) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)

\(+ PCV\) if to stabilize

Everything similar to Table M but now that the entry ratios \(r^*\) are limited

\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{LLM}\)

\(b^{LLM} = e - (C - 1)\mathrm{E}[A] + CI^{LLM}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped

\(\phi^{LLM}(r_H^*) - \phi^{LLM}(r_G^*) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A^*]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A^*]T}\)

\(r_G^* - r_H^* = \dfrac{G - H}{C\mathrm{E}[A^*]T}\)

Constructing Table L

Table L charge includes the charge for per occurrence limit

Implicit charge for the occ limit \(\in \{0, k\}\)

Table L uses different premium size groups and occurrence limits

The LER was derived from the combination of all premium groups

CA version of Table M shows a higher charge than the CW average due to higher variation in the CA rates

\(\phi^L(r) - \phi(r)\) shows in the data was smaller than the LER implies due to overlap between occ limit and agg limit

Main Difference from the M and LLM

Entry ratios = \(\dfrac{A^*}{\mathrm{E}[A]} = \dfrac{\text{Actual Limited Loss}}{\text{Expected }\textbf{Unlimited}\text{ Loss}}\)

2013 Q15b

If Standard Premium \(\neq\) \(\forall\) risk \(\Rightarrow\) Create Tbl L with P instead of # of risk

2008 Q32

Understanding Table L

alt text

Note that the y axis is \(Y = \dfrac{A^*}{\mathrm{E}[A]}\) and the CDF of interest is \(F^*(Y)\). However, the x axis of the graph is of \(F(Y)\)

Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)

Area under \(F(Y) = 1\) same as Table M \(\Rightarrow\) Area under \(F^*(Y) = \mathrm{E}[Y] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]} = 1 - k\)

\(\phi^L(r) = \int\limits_r^{\infty} (y-r)dF^*(y) + k\) Remember the k

\(\psi^L(r) = \int\limits_0^{\r} (r-y)dF^*(y)\)

Key properties all the same except:

\(\phi^L(\infty) = k\)

Net Insurance Charge - Table L

\(R^* = (b^L + C{\color{blue}{L^*}}) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)

Everything same as Table M but now that we use \(I^L\)

\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{L}\)

\(b^{L} = e - (C - 1)\mathrm{E}[A] + CI^{L}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped

\(\phi^{L}(r_H) - \phi^{L}(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)

\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)

2001 Q34

Table L vs NCCI

NCCI uses the ICRLL to \(\approx\) LLM

Table L Advantages:

Table L Disadvantages:

Example of Table L, LLM, and M

See Manual for example

Using NCCI Manual

Some of the important sections below. Cancellation provision is not on the syllabus

Expected Loss Group Tables (Pt 4-B)

Expected Loss Group = Risk Size Groups

To determine the ELG:

  1. Adjusted Expected Losses \(= \mathrm{E}[A] \times \text{State HG Differential} \times \dfrac{1 + 0.8k}{1-k}\)
    • \(k = LER\)
    • \(\dfrac{1 + 0.8k}{1-k}\) is the ICRLL procedure
      • LLM \(\approx\) Table M by adjusting the Adjusted Expected Losses to be for a ELG
      • Use LLM balance equation instead if using this
  2. Look up ELG with the Adjusted Expected Losses
    • Match the policy effective date with table effective date

\(\mathrm{E}[L]\) increase over time due to inflation. The ELG tables are updated so the curves so NCCI doesn’t have to update the curve. They’ll just push risk to different ELG.

Estimate portion of risk moving from ELG X to Y: \(\dfrac{\text{Portion of ELG X in new ELG Y}}{\text{Size of old ELG X}}\)

Table M (Pt 4-C)

Each column is for different ELG

For low entry ratios, Table M savings are shown as well

Expense Ratio Tables (Pt 4-D)

Not tested since 2000

This is for the \(e\) in \(b = e - (C - 1)\mathrm{E}[A] + CI\)

Type A = Stock; Type B = non-Stock

Shows ELR, premium discount ranges and tax multipliers

AK Special Rating Values

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